Poisson Equation

Solving the Poisson equation in a dielectric is:

$$ \epsilon \int \nabla \psi \cdot \partial r = 0 $$

which is equivalent to solving

$$ \epsilon \int \vec{E} \cdot \partial \vec{s} = 0 $$

In DEVSIM, the surface integral is performed by specifying an equation in the region with an edge model which is the negative gradient of the potential, $$\psi$$. An example implementation is located in

def CreateOxidePotentialOnly(device, region, update_type="default"): '''   Create electric field model in oxide Creates Potential solution variable if not available ''' if not InNodeModelList(device, region, "Potential"): print "Creating Node Solution Potential" CreateSolution(device, region, "Potential")

efield="(Potential@n0 - Potential@n1)*EdgeInverseLength" # this needs to remove derivatives w.r.t. independents CreateEdgeModel(device, region, "ElectricField", efield) CreateEdgeModelDerivatives(device, region, "ElectricField", efield, "Potential") dfield="Permittivity*ElectricField" CreateEdgeModel(device, region, "PotentialEdgeFlux", dfield) CreateEdgeModelDerivatives(device, region, "PotentialEdgeFlux", dfield, "Potential") equation(device=device, region=region, name="PotentialEquation", variable_name="Potential",     edge_model="PotentialEdgeFlux", variable_update=update_type)